Factoring. 472 Chapter 9 Factoring

472 Chapter 9 Factoring

Factoring

• factored form (p. 475) • factoring by grouping (p. 482) • prime polynomial (p. 497) • difference of squares (p. 501) • perfect square trinomials (p. 508)

Key Vocabulary

The factoring of polynomials can be used to solve a variety of real-world problems and lays the foundation for the further study of polynomial equations. Factoring is used to solve problems involving vertical motion. For example, the height h in feet of a dolphin that jumps out of the water traveling at 20 feet per second is modeled by a polynomial equation. Factoring can be used to determine how long the dolphin is in the air. You will learn how to solve polynomial equations in Lesson 9-2.

• Lesson 9-1 Find the prime factorizations of integers and monomials.

• Lesson 9-1 Find the greatest common factors (GCF) for sets of integers and monomials. • Lessons 9-2 through 9-6 Factor polynomials. • Lessons 9-2 through 9-6 Use the Zero Product

Chapter 9 Factoring 473 Make this Foldable to help you organize your notes on

factoring. Begin with a sheet of plain 81₂"by 11" paper.

Label each tab as shown. 9-1 9-2 9-3 9-4 9-5 9-6 F a c t o r i n g Open. Cut short side along

folds to make tabs.

Open. Fold lengthwise, leaving a tab on the right. 1 2"

Fold in thirds and then in half along

the width.

Reading and Writing As you read and study the chapter, write notes and examples for each lesson under its tab.

Prerequisite Skills

To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 9.

For Lessons 9-2 through 9-6 Distributive Property

Rewrite each expression using the Distributive Property. Then simplify.

(For review, see Lesson 1-5.)

1. 3(4x) 2. a(a5) 3. 7(n2_{3n1) 4. 6y(3y5y}2_y3₎

For Lessons 9-3 and 9-4 Multiplying Binomials

Find each product. (For review, see Lesson 8-7.)

5. (x4)(x7) 6. (3n4)(n5) 7. (6a2b)(9ab) 8. (x8y)(2x12y)

For Lessons 9-5 and 9-6 Special Products

Find each product. (For review, see Lesson 8-8.)

9. (y9)2 _{10. (3a2)}2 _{11. (n5)(n5) 12. (6p7q)(6p7q)}

For Lesson 9-6 Square Roots

Find each square root. (For review, see Lesson 2-7.) 13.

121

14.

0.0064

15.

2 3 5 6

16.

9 8 8

Fold in Sixths Cut Fold Again Label

PRIME FACTORIZATION

Recall that when two or more numbers are multiplied, each number is a factor of the product. Some numbers, like 18, can be expressed as the product of different pairs of whole numbers. This can be shown geometrically. Consider all of the possible rectangles with whole number dimensions that have areas of 18 square units.

The number 18 has 6 factors, 1, 2, 3, 6, 9, and 18. Whole numbers greater than 1 can be classified by their number of factors.

Vocabulary

• prime number • composite number • prime factorization • factored form

• greatest common factor (GCF)

Factors and Greatest

Common Factors

474 Chapter 9 Factoring

are prime numbers related to the

search for extraterrestrial life?

are prime numbers related to the

search for extraterrestrial life?

Classify Numbers as Prime or Composite

Factor each number. Then classify each number as prime or composite. a. 36

To find the factors of 36, list all pairs of whole numbers whose product is 36.

1 36 2 18 3 12 4 9 6 6

Therefore, the factors of 36, in increasing order, are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Since 36 has more than two factors, it is a composite number.

Example

1

Example

1

1 18

2 9 3 6

Prime and Composite Numbers

Words Examples

A whole number, greater than 1, whose only factors

are 1 and itself, is called a . 2, 3, 5, 7, 11, 13, 17, 19

A whole number, greater than 1, that has more than

two factors is called a . 4, 6, 8, 9, 10, 12, 14, 15, 16, 18

0 and 1 are neither prime nor composite.

composite number prime number

Study Tip

Listing Factors

Notice that in Example 1, 6 is listed as a factor of 36 only once.

• Find prime factorizations of integers and monomials.

• Find the greatest common factors of integers and monomials.

In the search for extraterrestrial life, scientists listen to radio signals coming from faraway galaxies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of coming from some natural phenomenon? What if that signal began with a series of beeps in a pattern comprised of the first 30 prime numbers (“beep-beep,” “beep-beep-beep,” and so on)?

Study Tip

Lesson 9-1 Factors and Greatest Common Factors 475

When a whole number is expressed as the product of factors that are all prime numbers, the expression is called the prime factorization of the number.

A negative integer is factored completely when it is expressed as the product of 1 and prime numbers.

www.algebra1.com/extra_examples

b. 23

The only whole numbers that can be multiplied together to get 23 are 1 and 23. Therefore, the factors of 23 are 1 and 23. Since the only factors of 23 are 1 and itself, 23 is a prime number.

Prime Factorization of a Positive Integer

Find the prime factorization of 90.

Method 1

90 245 The least prime factor of 90 is 2.

2315 The least prime factor of 45 is 3.

2335 The least prime factor of 15 is 3.

All of the factors in the last row are prime. Thus, the prime factorization of 90 is 2335.

Method 2

Use a factor tree. 90

9 10 90 910

3325 9 33 and 10 25

All of the factors in the last branch of the factor tree are prime. Thus, the prime factorization of 90 is 2335 or 232_5.

Usually the factors are ordered from the least prime factor to the greatest.

Example

2

Example

2

Prime Factorization of a Negative Integer

Find the prime factorization of 140.

140 1140 Express 140 as 1 times 140. 1270 140 2 70

12710 70 7 10 12725 10 2 5

Thus, the prime factorization of 140 is 1 2 2 5 7 or 1 225 7.

Example

3

Example

3

A monomial is in when it is expressed as the product of prime numbers and variables and no variable has an exponent greater than 1.

factored form

Study Tip

Unique

Factorization

Theorem

The prime factorization of every number is unique except for the order in which the factors are written.

Prime Numbers

Before deciding that a number is prime, try dividing it by all of the prime numbers that are less than the square root of that number.

• The GCF of two or more integers is the product of the prime factors common to the integers.

• The GCF of two or more monomials is the product of their common factors when each monomial is in factored form.

• If two or more integers or monomials have a GCF of 1, then the integers or monomials are said to be relatively prime.

Greatest Common Factor (GCF)

GREATEST COMMON FACTOR

Two or more numbers may have some common prime factors. Consider the prime factorization of 48 and 60.

48 2 2 2 2 3 Factor each number.

60 2 2 3 5 Circle the common prime factors.

The integers 48 and 60 have two 2s and one 3 as common prime factors. The product of these common prime factors, 2 2 3 or 12, is called the

of 48 and 60. The GCF is the greatest number that is a factor of both original numbers.

(GCF)

greatest common factor

476 Chapter 9 Factoring

Prime Factorization of a Monomial

Factor each monomial completely.

a. 12a2_b3

12a2b32 6 aabbb 12 2 6, a2_{aa, and b}3_bbb

2 2 3 aabbb 6 2 3

Thus, 12a2b3in factored form is 2 2 3 aabbb.

b. 66pq2

66pq2 1 66 pqq Express 66 as 1 times 66. 1 2 33 pqq 66 2 33

1 2 3 11 pqq 33 3 11

Thus, 66pq2in factored form is 1 2 3 11 pqq.

Example

4

Example

4

GCF of a Set of Monomials

Find the GCF of each set of monomials.

a. 15 and 16

15 3 5 Factor each number.

16 2 2 2 2 Circle the common prime factors, if any.

There are no common prime factors, so the GCF of 15 and 16 is 1. This means that 15 and 16 are relatively prime.

b. 36x2y and 54xy2z

36x2_{y 2 2 3 3 x x y Factor each number.}

54xy2_{z 2 3 3 3 x y y z Circle the common prime factors.} The GCF of 36x2_{y and 54xy}2_{z is 2 3 3 xy or 18xy.}

Example

5

Example

5

Alternative

Method

You can also find the greatest common factor by listing the factors of each number and finding which of the common factors is the greatest. Consider Example 5a. 15: 1, 3, 5, 15 16: 1, 2, 4, 8, 16 The only common factor, and therefore, the greatest common factor, is 1.

Practice and Apply

Practice and Apply

1. Determinewhether the following statement is true or false. If false, provide a counterexample.

All prime numbers are odd.

2. Explainwhat it means for two numbers to be relatively prime.

3. OPEN ENDED Name two monomials whose GCF is 5x2_.

Find the factors of each number. Then classify each number as prime or composite.

4. 8 5. 17 6. 112

Find the prime factorization of each integer.

7. 45 8. 32 9. 150

Factor each monomial completely.

10. 4p2 _{11. 39b}3_c2 _{12. 100x}3_yz2

Find the GCF of each set of monomials.

13. 10, 15 14. 18xy, 36y2 _{15. 54, 63, 180}

16. 25n, 21m 17. 12a2_{b, 90a}2_b2_{c 18. 15r}2_{, 35s}2_{, 70rs}

19. GARDENING Ashley is planting 120 tomato plants in her garden. In what ways can she arrange them so that she has the same number of plants in each row, at least 5 rows of plants, and at least 5 plants in each row?

Concept Check

Guided Practice

Application

Lesson 9-1 Factors and Greatest Common Factors 477

www.algebra1.com/self_check_quiz

Use Factors

GEOMETRY The area of a rectangle is 28 square inches. If the length and width are both whole numbers, what is the maximum perimeter of the rectangle?

Find the factors of 28, and draw rectangles with each length and width. Then find each perimeter.

The factors of 28 are 1, 2, 4, 7, 14, and 28.

The greatest perimeter is 58 inches. The rectangle with this perimeter has a length of 28 inches and a width of 1 inch.

Example

6

Example

6

P 1 28 1 28 or 58 1 28 14 2 P 2 14 2 14 or 32 7 4 P 4 7 4 7 or 22

GUIDED PRACTICE KEY

Find the factors of each number. Then classify each number as prime or composite.

20. 19 21. 25 22. 80 23. 61

478 Chapter 9 Factoring

Homework Help

For See Exercises Examples 20–27, 62, 1 65, 66 32–39 2, 3 40–47 4 48–61, 5 63, 64 28–31, 67 6

Extra Practice

See page 839.

Marching Bands

Drum Corps International (DCI) is a nonprofit youth organization serving junior drum and bugle corps around the world. Members of these marching bands range from 14 to 21 years of age.

Source:www.dci.org

GEOMETRY For Exercises 28 and 29, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers. 28. What is the minimum perimeter of the rectangle? Explain your reasoning.

29. What is the maximum perimeter of the rectangle? Explain your reasoning.

COOKIES For Exercises 30 and 31, use the following information.

A bakery packages cookies in two sizes of boxes, one with 18 cookies and the other with 24 cookies. A small number of cookies are to be wrapped in cellophane before they are placed in a box. To save money, the bakery will use the same size

cellophane packages for each box.

30. How many cookies should the bakery place in each cellophane package to maximize the number of cookies in each package?

31. How many cellophane packages will go in each size box?

Find the prime factorization of each integer.

32. 39 33. 98 34. 117 35. 102

36. 115 37. 180 38. 360 39. 462

Factor each monomial completely.

40. 66d4 _{41. 85x}2_y2 _{42. 49a}3_b2 _{43. 50gh}

44. 128pq2 _{45. 243n}3_{m 46. 183xyz}3 _{47. 169a}2_bc2

Find the GCF of each set of monomials.

48. 27, 72 49. 18, 35 50. 32, 48

51. 84, 70 52. 16, 20, 64 53. 42, 63, 105

54. 15a, 28b2 _{55. 24d}2_{, 30c}2_{d 56. 20gh, 36g}2_h2

57. 21p2_{q, 32r}2_{t 58. 18x, 30xy, 54y 59. 28a}2_{, 63a}3_b2_{, 91b}3

60. 14m2_n2_{, 18mn, 2m}2_n3 _{61. 80a}2_{b, 96a}2_b3_{, 128a}2_b2

62. NUMBER THEORY Twin primes are two consecutive odd numbers that are

prime. The first pair of twin primes is 3 and 5. List the next five pairs of twin primes.

MARCHING BANDS For Exercises 63 and 64, use the following information.

Central High’s marching band has 75 members, and the band from Northeast High has 90 members. During the halftime show, the bands plan to march into the stadium from opposite ends using formations with the same number of rows.

63. If the bands want to match up in the center of the field, what is the maximum number of rows?

64. How many band members will be in each row after the bands are combined?

NUMBER THEORY For Exercises 65 and 66, use the following information.

One way of generating prime numbers is to use the formula 2p_{1, where p is a} prime number. Primes found using this method are called Mersenne primes. For example, when p2, 22_{1 3. The first Mersenne prime is 3.}

65. Find the next two Mersenne primes.

66. Will this formula generate all possible prime numbers? Explain your reasoning.

Online Research

Data Update What is the greatest known prime

Finding the GCF of distances will help you make a scale model of the solar system. Visit

www.algebra1.com/ webquestto continue work on your WebQuest project.

Lesson 9-1 Factors and Greatest Common Factors 479

Getting Ready for

the Next Lesson

Maintain Your Skills

Maintain Your Skills

67. GEOMETRY The area of a triangle is 20 square centimeters. What are possible whole-number dimensions for the base and height of the triangle?

68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are natural numbers. Make a valid argument to explain why each assertion is true or provide a counterexample to show that an assertion is false.

a. 6 must be a factor of a or of b.

b. 3 must be a factor of a or of b.

c. 3 must be a factor of a and of b.

69. Answer the question that was posed at the beginning of the lesson.

How are prime numbers related to the search for extraterrestrial life?

Include the following in your answer:

• a list of the first 30 prime numbers and an explanation of how you found them, and

• an explanation of why a signal of this kind might indicate that an extraterrestrial message is to follow.

70. Miko claims that there are at least four ways to design a 120-square-foot rectangular space that can be tiled with 1-foot by 1-foot tiles. Which statement best describes this claim?

Her claim is false because 120 is a prime number. Her claim is false because 120 is not a perfect square. Her claim is true because 240 is a multiple of 120. Her claim is true because 120 has at least eight factors.

71. Suppose Ψ_xis defined as the largest prime factor of x. For which of the following values of x would Ψ_xhave the greatest value?

53 B 74 C 99 D 117 A D C B A WRITING IN MATH

Find each product. (Lessons 8-7 and 8-8)

72. (2x1)2 _{73. (3a5)(3a5) 74. (7p}2_4)(7p2₄₎

75. (6r7)(2r5) 76. (10hk)(2h5k) 77. (b4)(b2_3b18)

Find the value of r so that the line that passes through the given points has the given slope. (Lesson 5-1)

78. (1, 2), (2, r), m3 79. (5, 9), (r, 6), m 3

5

80. RETAIL SALES A department store buys clothing at wholesale prices and then marks the clothing up 25% to sell at retail price to customers. If the retail price of a jacket is $79, what was the wholesale price? (Lesson 3-7)

PREREQUISITE SKILL Use the Distributive Property to rewrite each expression.

(To review the Distributive Property, see Lesson 1-5.)

81. 5(2x8) 82. a(3a1) 83. 2g(3g4)

84. 4y(3y6) 85. 7b7c 86. 2x3x

Standardized

Test Practice

480Investigating Slope-Intercept Form 480 Chapter 9 Factoring

A Preview of Lesson 9-2

Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials.

Factoring Using the

Distributive Property

Model and Analyze

Use algebra tiles to factor each binomial.

1. 2x10 2. 6x8 3. 5x2_{2x 4. 93x}

Tell whether each binomial can be factored. Justify your answer with a drawing.

5. 4x10 6. 3x7 7. x2_{2x 8. 2x}2₃

9. MAKE A CONJECTURE Write a paragraph that explains how you can use algebra tiles to determine whether a binomial can be factored. Include an example of one binomial that can be factored and one that cannot.

Activity 1

Use algebra tiles to factor 3x 6.

Model the polynomial 3x6. Arrange the tiles into a rectangle. The total area of the rectangle

represents the product, and its length and width represent the factors.

The rectangle has a width of 3 and a length of x2. So, 3x63(x2). 3 x 2 x 1 1 x 1 1 x 1 1 1 1 1 1 1 1 x x x

Model the polynomial x2_{4x. Arrange the tiles into a rectangle.}

The rectangle has a width of x and a length of x4. So, x2_{4x x(x4).}

x x2 xxxx

x 4

x x x x x2

FACTOR BY USING THE DISTRIBUTIVE PROPERTY

In Chapter 8, you used the Distributive Property to multiply a polynomial by a monomial.

2a(6a8) 2a(6a) 2a(8)

12a2_16a

You can reverse this process to express a polynomial as the product of a monomial factor and a polynomial factor.

12a2_{16a2a(6a) 2a(8)}

2a(6a8) Thus, a factored form of 12a2_{16a is 2a(6a8).}

a polynomial means to find its completely factored form. The expression 2a(6a8) is not completely factored since 6a8 can be factored as 2(3a4).

Factoring

Factoring Using the

Distributive Property

Lesson 9-2 Factoring Using the Distributive Property 481

can you determine how long a

baseball will remain in the air?

can you determine how long a

baseball will remain in the air?

Use the Distributive Property

Use the Distributive Property to factor each polynomial. a. 12a2_16a

First, find the GCF of 12a2and 16a. 12a2_{22 3 aa Factor each number.}

16a2 2 2 2 a Circle the common prime factors. GCF: 2 •2 •a or 4a

Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.

12a2_{16a4a(3a) 4a(2 2) Rewrite each term using the GCF.}

4a(3a) 4a(4) Simplify remaining factors.

4a(3a4) Distributive Property

Thus, the completely factored form of 12a216a is 4a(3a4).

Example

1

Example

1

Nolan Ryan, the greatest strike-out pitcher in the history of baseball, had a fastball clocked at 98 miles per hour or about 151 feet per second. If he threw a ball directly upward with the same velocity, the height h of the ball in feet above the point at which he released it could be modeled by the formula h151t16t2_{, where}

t is the time in seconds. You can use factoring

and the Zero Product Property to determine how long the ball would remain in the air before returning to his glove.

• Factor polynomials by using the Distributive Property.

• Solve quadratic equations of the form ax2_bx0.

Vocabulary

• factoring

• factoring by grouping

Look Back

To review the Distributive Property, see Lesson 1-5.

The Distributive Property can also be used to factor some polynomials having four or more terms. This method is called because pairs of terms are grouped together and factored. The Distributive Property is then applied a second time to factor a common binomial factor.

factoring by grouping

482 Chapter 9 Factoring

Use Grouping

Factor 4ab8b3a6.

4ab8b3a6

(4ab8b) (3a 6) Group terms with common factors.

4b(a2) 3(a2) Factor the GCF from each grouping.

(a 2)(4b3) Distributive Property

CHECK Use the FOIL method.

F O I L (a2)(4b3)(a)(4b) (a)(3) (2)(4b) (2)(3) 4ab 3a 8b 6

Example

2

Example

2

Factoring by

Grouping

Sometimes you can group terms in more than one way when factoring a polynomial. For example, the polynomial in Example 2 could have been factored in the following way. 4ab8b3a6

(4ab3a) (8b6)

a(4b3) 2(4b3)

(4b3)(a2) Notice that this result is the same as in Example 2.

Study Tip

Factoring

Trinomials

Since the order in which factors are multiplied does not affect the product, (5x3)(y7) is also a correct factoring of 35x5xy3y21.

Study Tip

Factoring by Grouping

• Words A polynomial can be factored by grouping if all of the following

situations exist.

• There are four or more terms.

• Terms with common factors can be grouped together.

• The two common factors are identical or are additive inverses of each other.

• Symbols axbxaybyx(ab) y(ab)

(ab)(xy) b. 18cd2_12c2_d9cd

18cd2_{233cdd Factor each number.}

12c2_{d223ccd Circle the common prime factors.} 9cd33cd

GCF: 3cd or 3cd

18cd2_12c2_{d9cd3cd(6d) 3cd(4c) 3cd(3) Rewrite each term using the GCF.}

3cd(6d4c3) Distributive Property

Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For example, 7 y and y7 are additive inverses because their sum is 0. By rewriting 7 y in the factored form 1(y7), factoring by grouping is made possible in the following example.

Use the Additive Inverse Property

Factor 35x5xy3y21.

35x5xy3y21 (35x5xy) (3y21) Group terms with common factors.

5x(7 y) 3(y7) Factor the GCF from each grouping.

5x(1)(y7) 3(y7) 7 y 1(y7)

5x(y7) 3(y7) 5x(1) 5x

(y 7)(5x3) Distributive Property

Example

3

Zero Product Property

• Words If the product of two factors is 0, then at least one of the factors

must be 0.

• Symbols For any real numbers a and b, if ab0, then either a0, b0, or both a and b equal zero.

Lesson 9-2 Factoring Using the Distributive Property 483

SOLVE EQUATIONS BY FACTORING

Some equations can be solved by factoring. Consider the following products.

6(0)0 0(3) 0 (5 5)(0) 0 2(3 3) 0

Notice that in each case, at least one of the factors is zero. These examples illustrate the Zero Product Property.

Solve an Equation in Factored Form

Solve (d5)(3d4) 0. Then check the solutions.

If (d5)(3d4) 0, then according to the Zero Product Property either

d5 0 or 3d4 0.

(d5)(3d4) 0 Original equation

d5 0 or 3d4 0 Set each factor equal to zero.

d5 3d 4 Solve each equation.

d 4

3 The solution set is

5, 4

3

.

CHECK Substitute 5 and 4

3for d in the original equation.

(d5)(3d4)0 (d5)(3d4) 0 (55)[3(5)4] 0

4 3 5

3

4 3

4

0 (0)(19) 0

1 3 9

(0) 0 0 0 0 0

Example

4

Example

4

Common

Misconception

You may be tempted to try to solve the equation in Example 5 by dividing each side of the equation by x. Remember, however, that xis an unknown quantity. If you divide by x, you may actually be dividing by zero, which is undefined.

Study Tip

_{Solve an Equation by Factoring}

Solve x2_{7x. Then check the solutions.}

Write the equation so that it is of the form ab0.

x27x Original equation

x2_{7x0 Subtract 7x from each side.}

x(x 7) 0 Factor the GCF of x2_{and 7x, which is x.}

x0 or x7 0 Zero Product Property

x7 Solve each equation.

The solution set is {0, 7}. Check by substituting 0 and 7 for x in the original equation.

Example

5

Example

5

If an equation can be written in the form ab0, then the Zero Product Property can be applied to solve that equation.

484 Chapter 9 Factoring

Practice and Apply

Practice and Apply

1. Write4x2_{12x as a product of factors in three different ways. Then decide which} of the three is the completely factored form. Explain your reasoning.

2. OPEN ENDED Give an example of the type of equation that can be solved by using the Zero Product Property.

3. Explainwhy (x2)(x4)0 cannot be solved by dividing each side by x2.

Factor each polynomial.

4. 9x2_{36x 5. 16xz40xz}2

6. 24m2_np2_36m2_n2_{p 7. 2a}3_b2_8ab16a2_b3

8. 5y2_{15y4y12 9. 5c 10c}2_2d4cd

Solve each equation. Check your solutions.

10. h(h5) 0 11. (n4)(n2) 0 12. 5m3m2

PHYSICAL SCIENCE For Exercises 13–15, use the information below and in the graphic.

A flare is launched from a life raft. The height h of the flare in feet above the sea is modeled by the formula h100t16t2_{, where t is the time in} seconds after the flare is launched.

13. At what height is the flare when it returns to the sea?

14. Let h0 in the equation h100t 16t2_and solve for t.

15. How many seconds will it take for the flare to return to the sea? Explain your reasoning.

h 100t 16t2

100 ft /s

h 0

Factor each polynomial.

16. 5x30y 17. 16a4b 18. a5_ba

19. x3_y2_{x 20. 21cd3d 21. 14gh18h}

22. 15a2_{y30ay 23. 8bc}2_{24bc 24. 12x}2_y2_z40xy3_z2

25. 18a2_bc2_48abc3 _{26. aa}2_b2_a3_b3 _{27. 15x}2_y2_{25xy x}

28. 12ax3_20bx2_{32cx 29. 3p}3_q9pq2_{36pq 30. x}2_2x3x6

31. x2_{5x7x35 32. 4x}2 _{14x6x21 33. 12y}2_{9y 8y6}

34. 6a2_{15a8a20 35. 18x}2_30x3x5

36. 4ax3ay4bx3by 37. 2my7x7m2xy

38. 8ax6x12a9 39. 10x2_14xy15x21y

GEOMETRY For Exercises 40 and 41, use the following information.

A quadrilateral has 4 sides and 2 diagonals. A pentagon has 5 sides and 5 diagonals. You can use 1

2n2 3

2n to find the number of diagonals in a polygon with n sides.

40. Write this expression in factored form.

41. Find the number of diagonals in a decagon (10-sided polygon).

Homework Help

For See Exercises Examples 16–29, 1 40–47 30–39 2, 3 48–61 4, 5

Extra Practice

See page 840.

Concept Check

Guided Practice

Application

Lesson 9-2 Factoring Using the Distributive Property 485 SOFTBALL For Exercises 42 and 43, use the following information.

Albertina is scheduling the games for a softball league. To find the number of games she needs to schedule, she uses the equation g1

2n2 1

2n, where g represents the number of games needed for each team to play each other team exactly once and n represents the number of teams.

42. Write this equation in factored form.

43. How many games are needed for 7 teams to play each other exactly 3 times?

GEOMETRY Write an expression in factored form for the area of each shaded region.

44. 45.

GEOMETRY Find an expression for the area of a square with the given perimeter. 46. P12x20y in. 47. P36a16b cm

Solve each equation. Check your solutions.

48. x(x24) 0 49. a(a16) 0 50. (q4)(3q15) 0 51. (3y9)(y7) 0 52. (2b3)(3b8) 0 53. (4n5)(3n7) 0 54. 3z2_{12z0 55. 7d}2_35d0 56. 2x2_{5x 57. 7x}2_6x 58. 6x2 _{4x 59. 20x}2 _15x

60. MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after

t seconds is given by the formula h20t16t2. How long is the dolphin in the air before returning to the water?

61. BASEBALL Malik popped a ball straight up with an initial upward velocity of 45 feet per second. The height h, in feet, of the ball above the ground is modeled by the equation h2 48t16t2_{. How long was the ball in the air if the} catcher catches the ball when it is 2 feet above the ground?

62. CRITICAL THINKING Factor axyaxbyaybxbxy.

63. Answer the question that was posed at the beginning of the lesson.

How can you determine how long a baseball will remain in the air?

Include the following in your answer:

• an explanation of how to use factoring and the Zero Product Property to find how long the ball would be in the air, and

• an interpretation of each solution in the context of the problem.

WRITING IN MATH r r 2 2 2 2 a b www.algebra1.com/self_check_quiz

Marine Biologist

Marine biologists study factors that affect organisms living in and near the ocean.

Online Research

For information about a career as a marine biologist, visit:

www.algebra1.com/ careers

486 Chapter 9 Factoring

P

ractice Quiz 1

P

ractice Quiz 1

1. Find the factors of 225. Then classify the number as prime or composite. (Lesson 9-1)

2. Find the prime factorization of 320. (Lesson 9-1)

3. Factor 78a2_bc3_{completely. (Lesson 9-1)}

4. Find the GCF of 54x3_{, 42x}2_{y, and 30xy}2_{. (Lesson 9-1)}

Factor each polynomial. (Lesson 9-2)

5. 4xy2_{xy 6. 32a}2_b40b3_8a2_b2 _{7. 6py16p15y40}

Solve each equation. Check your solutions. (Lesson 9-2)

8. (8n5)(n4) 0 9. 9x2_{27x0 10. 10x}2 _3x

64. The total number of feet in x yards, y feet, and z inches is

3xy₁z₂. 12(xyz). x3y36z. 3 x 6 1 y 2 z.

65. QUANTITATIVE COMPARISON Compare the quantity in Column A and the quantity in Column B. Then determine whether:

the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or

the relationship cannot be determined from the information given. D C B A D C B A

Standardized

Test Practice

Maintain Your Skills

Maintain Your Skills

Factor each number. Then classify each number as prime or composite. (Lesson 9-1)

66. 123 67. 300 68. 67

Find each product. (Lesson 8-8)

69. (4s3₃₎2 _{70. (2p5q)(2p5q) 71. (3k8)(3k8)}

Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)

72. s s 4 7 73. 1 1 8 2 x x 3 2 y y 4 1 74. 1 3 7 4 ( p p 7 3 q q 2 r r 1 5 )2

75. FINANCE Michael uses at most 60% of his annual FlynnCo stock dividend to purchase more shares of FlynnCo stock. If his dividend last year was $885 and FlynnCo stock is selling for $14 per share, what is the greatest number of shares that he can purchase? (Lesson 6-2)

PREREQUISITE SKILL Find each product.

(To review multiplying polynomials, see Lesson 8-7.)

76. (n8)(n3) 77. (x4)(x 5) 78. (b10)(b7)

79. (3a1)(6a4) 80. (5p2)(9p3) 81. (2y5)(4y3)

Mixed Review

the negative solution of the negative solution of (a2)(a5) 0 (b6)(b1) 0

Column A Column B

Getting Ready for

the Next Lesson

Algebra Activity Factoring Trinomials 487

A Preview of Lesson 9-3

Factoring Trinomials

You can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangle formed by algebra tiles, then the rectangle’s length and width are factors of the area.

(continued on the next page)

Activity 1

Use algebra tiles to factor x2 _{6x 5.} Model the polynomial x2_6x5.

Place the x2_{tile at the corner of the} product mat. Arrange the 1 tiles into a rectangular array. Because 5 is prime, the 5 tiles can be arranged in a rectangle in one way, a 1-by-5 rectangle.

Complete the rectangle with the x tiles. The rectangle has a width of x1 and a length of x5. Therefore,

x2_{6x + 5 (x1)(x5).} x 1 x 5 1 1 1 1 1 x2 x x x x x x 1 1 1 1 1 x2 x2 1 1 1 1 1 x x x x x x

Activity 2

Use algebra tiles to factor x2 _{7x 6.} Model the polynomial

x2_7x6.

Place the x2tile at the corner of the product mat. Arrange the 1 tiles into a rectangular array. Since 6 2 3, try a 2-by-3 rectangle. Try to complete the rectangle. Notice that there are two extra x tiles.

1 1 1 1 1 1 x2 x2 x x x x x 1 1 1 1 1 1 x x

488Investigating Slope-Intercept Form

Algebra Activity

Algebra Activity

488 Chapter 9 Factoring

Arrange the 1 tiles into a 1-by-6 rectangular array. This time you can complete the rectangle with the x tiles.

The rectangle has a width of x1 and a length of x6. Therefore,

x2_{7x6 (x1)(x6).} x 1 x 6 1 1 1 1 1 1 x2 x x x x x x x

Activity 3

Use algebra tiles to factor x2 _{2x 3.} Model the polynomial x22x3.

Place the x2tile at the corner of the product mat. Arrange the 1 tiles into a 1-by-3 rectangular array as shown.

Place the x tile as shown. Recall that you can add zero-pairs without changing the value of the polynomial. In this case, add a zero pair of x tiles.

The rectangle has a width of x1 and a length of x3. Therefore, x2_{2x3 (x1)(x3).} x2 xxx x 1 x 3 x2 111 111 xx x zero pair x2 111 x x x2 1 1 1

Model

Use algebra tiles to factor each trinomial.

1. x2_{4x3 2. x}2_{5x4 3. x}2_{x6 4. x}2_3x2

Factoring Trinomials:

x

2

bx

c

can factoring be used to find the dimensions of a garden?

can factoring be used to find the dimensions of a garden?

• Factor trinomials of the form x2_bxc.

• Solve equations of the form x2_bxc0.

Tamika has enough bricks to make a 30-foot border around the rectangular vegetable garden she is planting. The booklet she got from the nursery says that the plants will need a space of 54 square feet to grow. What should the dimensions of her garden be? To solve this problem, you need to find two numbers whose product is 54 and whose sum is 15, half the perimeter of the garden.

A 54 ft2

P 30 ft

Reading Math

A quadratic trinomialis a trinomial of degree 2. This means that the greatest exponent of the variable is 2.

Study Tip

Factoring x

2

bx

c

• Words To factor quadratic trinomials of the form x2_{bxc, find}

two integers, m and n, whose sum is equal to b and whose product is equal to c. Then write x2_{bxc using the pattern (xm)(xn).}

• Symbols x2_{bxc(xm)(xn) when mnb and mnc.}

• Example x2_{5x6(x2)(x3), since 235 and 236.}

FACTOR

x

2

bx

c

In Lesson 9-1, you learned that when two numbers are multiplied, each number is a factor of the product. Similarly, when two binomials are multiplied, each binomial is a factor of the product.

To factor some trinomials, you will use the pattern for multiplying two binomials. Study the following example.

F O I L

(x2)(x3)(xx)(x3)(x2)(23) Use the FOIL method.

x2_{3x2x6 Simplify.}

x2(32)x6 Distributive Property

x2_{5x6 Simplify.}

Observe the following pattern in this multiplication. (x2)(x3)x2(32)x(23) (xm)(xn)x2(nm)xmn

x2(mn)xmn

x2 _{bx c bmnand cmn}

Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n. This pattern can be used to factor quadratic trinomials of the form x2bxc.

To determine m and n, find the factors of c and use a guess-and-check strategy to find which pair of factors has a sum of b.

490 Chapter 9 Factoring

b and c Are Positive

Factor x2_6x8.

In this trinomial, b6 and c8. You need to find two numbers whose sum is 6 and whose product is 8. Make an organized list of the factors of 8, and look for the pair of factors whose sum is 6.

Factors of 8 Sum of Factors

1, 8 9

2, 4 6 The correct factors are 2 and 4.

x2_{6x8(xm)(xn) Write the pattern.}

(x2)(x4) m2 and n4

CHECK You can check this result by multiplying the two factors.

F O I L (x2)(x4)x24x2x8 FOIL method x26x8 Simplify.

Example

1

Example

1

Testing Factors

Once you find the correct factors, there is no need to test any other factors. Therefore, it is not necessary to test 4 and

4 in Example 2.

Study Tip

b Is Negative and c Is Positive

Factor x210x16.

In this trinomial, b 10 and c16. This means that mn is negative and mn is

positive. So m and n must both be negative. Therefore, make a list of the negative factors of 16, and look for the pair of factors whose sum is 10.

Factors of 16 Sum of Factors

1,16 17

2, 8 10

4, 4 8

The correct factors are 2 and 8.

x2_{10x16(xm)(xn) Write the pattern.}

(x2)(x8) m 2 and n 8 CHECK You can check this result by using a

graphing calculator. Graph yx2_10x16 and y(x2)(x8) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly.

[10, 10] scl: 1 by [10, 10] scl: 1

Example

2

Example

2

When factoring a trinomial where b is negative and c is positive, you can use what you know about the product of binomials to help narrow the list of possible factors.

You will find that keeping an organized list of the factors you have tested is particularly important when factoring a trinomial like x2_{x12, where the value} of c is negative.

Lesson 9-3 Factoring Trinomials: x2_{bxc 491}

SOLVE EQUATIONS BY FACTORING

Some equations of the form

x2_{bxc0 can be solved by factoring and then using the Zero Product Property.}

www.algebra1.com/extra_examples

b Is Positive and c Is Negative

Factor x2x12.

In this trinomial, b1 and c 12. This means that mn is positive and mn is

negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 12, where one factor of each pair is negative. Look for the pair of factors whose sum is 1.

Factors of 12 Sum of Factors

1,12 11

1, 12 11

2, 6 4

2, 6 4

3, 4 1

3, 4 1 The correct factors are 3 and 4.

x2_{x12(xm)(xn) Write the pattern.}

(x3)(x4) m 3 and n4

Example

3

Example

3

b Is Negative and c Is Negative

Factor x27x18.

Since b 7 and c 18, mn is negative and mn is negative. So either m or n

is negative, but not both.

Factors of 18 Sum of Factors

1,18 17

1, 18 17

2, 9 7 The correct factors are 2 and 9.

x27x18(xm)(xn) Write the pattern.

(x2)(x9) m2 and n 9

Example

4

Example

4

Solve an Equation by Factoring

Solve x25x6. Check your solutions.

x2_{5x6 Original equation}

x2_{5x60 Rewrite the equation so that one side equals 0.} (x1)(x6)0 Factor.

x10 or x60 Zero Product Property

x1 x 6 Solve each equation. The solution set is {1, 6}.

CHECK Substitute 1 and 6 for x in the original equation.

x2_{5x6 x}2_5x6 (1)2_{5(1)6 (6)}2₅₍₆₎₆ 66 66

Example

5

Example

5

Alternate Method

You can use the opposite of FOIL to factor trinomials. For instance, consider Example 3.

x2_x12

(x■)(x■) Try factor pairs of 12 until the sum of the products of the Inner and Outer terms is x.

Study Tip

Guided Practice

492 Chapter 9 Factoring

Solve a Real-World Problem by Factoring

YEARBOOK DESIGN A sponsor for

the school yearbook has asked that the length and width of a photo in their ad be increased by the same amount in order to double the area of the photo. If the photo was originally 12 centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be?

Explore Begin by making a diagram like the one shown above, labeling the appropriate dimensions.

Plan Let xthe amount added to each dimension of the photo. The new length times the new width equals the new area.

x12 x8 2(8)(12)

old area Solve (x12)(x8)2(8)(12) Write the equation.

x2_{20x96192 Multiply.}

x2_{20x960 Subtract 192 from each side.} (x24)(x4)0 Factor.

x240 or x40 Zero Product Property

x 24 x4 Solve each equation.

Examine The solution set is {24, 4}. Only 4 is a valid solution, since dimensions cannot be negative. Thus, the new length of the photo should be 412 or 16 centimeters, and the new width should be 48 or 12 centimeters.

12 8 x x

Example

6

Example

6

GUIDED PRACTICE KEY

Concept Check

1. Explainwhy, when factoring x26x9, it is not necessary to check the sum of the factor pairs 1 and 9 or 3 and 3.

2. OPEN ENDED Give an example of an equation that can be solved using the factoring techniques presented in this lesson. Then, solve your equation.

3. FIND THE ERROR Peter and Aleta are solving x22x15.

Who is correct? Explain your reasoning.

Factor each trinomial.

4. x2_{11x24 5. c}2_{3c2 6. n}2_13n48 7. p2_{2p35 8. 7227aa}2 _{9. x}2_4xy3y2 Aleta x2 + 2x = 15 x2 _{+ 2x - 15 = 0} (x - 3)(x + 5) = 0 x - 3 = 0 or x + 5 = 0 x = 3 x = -5 Peter x2_{+ 2x = 15} x (x + 2) = 15 x = 15 or x + 2 = 15 x = 13

Lesson 9-3 Factoring Trinomials: x2_{bxc 493}

Solve each equation. Check your solutions.

10. n2_{7n60 11. a}2_{5a360 12. p}2_19p420

13. y2_{9 10y 14. 9xx}2_{22 15. d}2_3d70

16. NUMBER THEORY Find two consecutive integers whose product is 156.

Application

www.algebra1.com/self_check_quiz

Practice and Apply

Practice and Apply

Factor each trinomial.

17. a28a15 18. x212x27 19. c212c35

20. y213y30 21. m222m21 22. d27d10

23. p217p72 24. g219g60 25. x26x7

26. b2b20 27. h23h40 28. n23n54

29. y2y42 30. z218z40 31. 726ww2

32. 3013xx2 33. a25ab4b2 34. x213xy36y2

GEOMETRY Find an expression for the perimeter of a rectangle with the given area.

35. areax224x81 36. areax213x90

Solve each equation. Check your solutions.

37. x216x280 38. b220b360 39. y24y120 40. d22d80 41. a23a280 42. g24g450 43. m219m480 44. n222n720 45. z2187z 46. h215 16h 47. 24k210k 48. x220x 49. c250 23c 50. y229y 54 51. 14pp251 52. x22x674 53. x2x5617x

54. SUPREME COURT When the Justices of the Supreme Court assemble to go on the Bench each day, each Justice shakes hands with each of the other Justices for a total of 36 handshakes. The total number of handshakes h possible for

n people is given by hn2₂n. Write and solve an equation to determine the number of Justices on the Supreme Court.

55. NUMBER THEORY Find two consecutive even integers whose product is 168.

56. GEOMETRY The triangle has an area of 40 square centimeters. Find the height h of the triangle.

CRITICAL THINKING Find all values of k so that each trinomial can be factored using integers.

57. x2_{kx19 58. x}2_kx14

59. x2_{8xk , k0 60. x}2_{5xk, k0}

RUGBY For Exercises 61 and 62, use the following information.

The length of a Rugby League field is 52 meters longer than its width w.

61. Write an expression for the area of the field.

62. The area of a Rugby League field is 8160 square meters. Find the dimensions of the field.

(2h 6) cm

h cm

Supreme Court

The “Conference handshake” has been a tradition since the late 19th century. Source:www.supremecourtus.gov

Homework Help

For See Exercises Examples 17–36 1–4 37–53 5 54–56, 6 61, 62

Extra Practice

See page 840.

Maintain Your Skills

Maintain Your Skills

Solve each equation. Check your solutions. (Lesson 9-2)

70. (x3)(2x5)0 71. b(7b4)0 72. 5y2 _9y

Find the GCF of each set of monomials. (Lesson 9-1)

73. 24, 36, 72 74. 9p2_q5_{, 21p}3_q3 _{75. 30x}4_y5_{, 20x}2_y7_{, 75x}3_y4

INTERNET For Exercises 76 and 77, use the graph at the right.

(Lessons 3-7 and 8-3)

76. Find the percent increase in the number of domain registrations from 1997 to 2000.

77. Use your answer from Exercise 76 to verify the claim that registrations grew more than 18-fold from 1997 to 2000 is correct.

PREREQUISITE SKILL Factor each polynomial.

(To review factoring by grouping, see Lesson 9-2.)

78. 3y2_{2y9y6 79. 3a}2_{2a12a8 80. 4x}2_3x8x6 81. 2p2_{6p7p21 82. 3b}2_{7b12b28 83. 4g}2_2g6g3 494 Chapter 9 Factoring

Graphing

Calculator

Mixed Review

63. Answer the question that was posed at the beginning of the lesson.

How can factoring be used to find the dimensions of a garden?

Include the following in your answer:

• a description of how you would find the dimensions of the garden, and • an explanation of how the process you used is related to the process used to

factor trinomials of the form x2_bxc.

64. Which is the factored form of x217x42?

(x1)(y42) (x2)(x21)

(x3)(x14) (x6)(x7)

65. GRID IN What is the positive solution of p213p300?

Use a graphing calculator to determine whether each factorization is correct. Write yes or no. If no, state the correct factorization.

66. x214x48(x6)(x8) 67. x216x105(x5)(x21) 68. x225x66(x33)(x2) 69. x211x210(x10)(x21) D C B A WRITING IN MATH

Standardized

Test Practice

Getting Ready for

the Next Lesson

USA TODAY Snapshots

®

By Cindy Hall and Bob Laird, USA TODAY Source: Network Solutions (VeriSign)

Number of domain

registrations climbs

During the past four years, .com, .net and .org domain registrations have grown more than 18-fold:

28.2 million 1997 1998 1999 2000 1.54 million 3.36 million 9 million

FACTOR

ax

2

_bx

_c

_{For trinomials of the form x}2_{bxc, the coefficient} of x2_{is 1. To factor trinomials of this form, you find the factors of c whose sum is b.} We can modify this approach to factor trinomials whose leading coefficient is not 1.

F O I L

(2x5)(3x1) 6x2_{2x15x5 Use the FOIL method.} 215 30

65 30 Observe the following pattern in this product.

6x2_{2x15x5 ax}2_mxnxc

6x2 _{17x 5 ax}2 _{bx c}

21517 and 21565 mnb and mnac

You can use this pattern and the method of factoring by grouping to factor 6x2_{17x5. Find two numbers, m and n, whose product is 65 or 30 and whose} sum is 17.

Factors of 30 Sum of Factors

1, 30 31

2, 15 17 The correct factors are 2 and 15.

6x2_{17x5 6x}2_{mxnx5 Write the pattern.}

6x2_{2x15x5 m2 and n15}

(6x2_{2x) (15x5) Group terms with common factors.}

2x(3x1) 5(3x1) Factor the GCF from each grouping.

(3x1)(2x5) 3x1 is the common factor. Therefore, 6x2_{17x5 (3x1)(2x 5).}

Factoring Trinomials:

ax

2

bx

c

Lesson 9-4 Factoring Trinomials: ax2_{bxc 495}

Vocabulary

• prime polynomial

• Factor trinomials of the form ax2_bxc.

• Solve equations of the form ax2_bxc0.

The factors of 2x27x6 are the dimensions of the rectangle formed by the algebra tiles shown below.

The process you use to form the rectangle is the same mental process you can use to factor this trinomial algebraically.

x2 1 1 1 1 1 1 x x2 x x x x x x

Look Back

To review factoring by grouping, see Lesson 9-2.

Study Tip

can algebra tiles be used to factor 2

x

2

₇

_x

_6?

can algebra tiles be used to factor 2

x

2

₇

_x

_6?